Slashdot videos: Now with more Slashdot!

View

Discuss

Share

We've improved Slashdot's video section; now you can view our video interviews, product close-ups and site visits with all the usual Slashdot options to comment, share, etc. No more walled garden! It's a work in progress -- we hope you'll check it out (Learn more about the recent updates).

Anti-Globalism suggests an article at Science News on the passing of Henri Cartan, one of the founding members of a strange and influential group of French mathematicians in the twentieth century. "In the 1930s, a group of young French mathematicians led an uprising that revolutionized mathematics. France had lost most of a generation in the First World War, so the emerging hotshots in mathematics had few elders to look up to. And when these radicals did look up, they didn't like what they saw. The practice of mathematics at the time was dry, scattered and muddled, they believed, in need of reinvention and invigoration... Using the nom de plume Nicolas Bourbaki (after a dead Napoleonic general), they wrote a series of textbooks laying out mathematics the right way. Though the young mathematicians started out only intending to write a good textbook for analysis..., they ended up creating dozens of volumes which formed a manifesto for a new philosophy of mathematics. The last of the founders of Bourbaki, Henri Cartan, died August 13 at age 104... Two of his students won the Fields medal..., one won the Nobel Prize in physics and another won the economics Nobel."

"Secret society" is a bit over the top! I always had the impression it had the feeling more of a running joke; from the article:

In the 1950s, Ralph Boas of Northwestern University wrote an article for the Encyclopaedia Britannica on Bourbaki, explaining that it was the pseudonym for a consortium of French mathematicians. The editors of the encyclopedia soon received a scalding letter signed by Nicolas Bourbaki himself, declaring that he would not allow anyone to question his right to exist. In revenge, Bourbaki began spreading the rumor that Ralph Boas himself didn't exist, and that B.O.A.S was an acronym of a group of American mathematicians.

Though it wasn't *just* a joke--they wrote a lot of very serious mathematics!

Bourbaki books are the most boring books you can buy. Avoid them at all costs. If you want to study mathematics, there are much better books.

Their motto is to never explain anything. These makes these books completely unreadable. Mathematics the right way ccroding to them is:--no example: examples are evil being that stray us from the true path of pure abstraction.--never mention any applications: are you nuts ? mathematics must remain pure. Applied mathematics are the spawn of the devil. If it serves some purpose, it's not mathematics anymore.--Don't draw anything. Drawings are tools of the devil. 2D domains and geometrical figures should only exists as pure abstraction.--The less explanations, the better: only idiots needs explanations.--Never rewrite a theorem for the sake of clarity: having 20 references to other theorems(usually in another volume) in a 5 lines proofis better for clarity (don't even write the name of the Theorem you refer to, a true mathematician knows them by volume and page number).

And better they add insult to the injury in the preface: "no prerequisite knowledge of math is needed to read this book". Yeah whatever.

Umm. I have a few texts from the late 1800s, and they are absolutly completely worse. Bourbaki's is the abstraction. ( I have their volume on abstract algebra, that I referred to while I was taking that graduate level course. Theirs is a terse work, much more accurate, and well though out. ) A few Springer-Verlag texts are worse also.

Indeed. And the much more literary style that was deemed acceptable before resulted not only in inaccuracy but in gross errors.

Bourbaki's work is an amazing feat, which nowadays can be appreciated maybe only with a considerable amount of historical perspective---mostly because it was extremely successful: it set (maybe by using an elaborate, laborious, hyperbole that is, among many other things, a display of love for the subjects treated) standards against which mathematical writing was (and is!) compared, if not jugded, and the student of today has the false impression that the textbooks he reads today are of the same kind as those that were read at all times, simply because he does not know history.

The effort spent in coming up with clear, precise definitions, detailed proofs, even with usable notation, is easy to disparage once one can enjoy its benefits.

There were no gross errors found by the Bourbaki group. This is why their horrible formal writing style died out: it increased the pain of writing without the gain of the previous waves of math formalization.

Bourbaki books are the most boring books you can buy. Avoid them at all costs.

This is a bit extreme. While they're certainly not the best books to begin learning a subject, they're great reference books. They're well written, (generally) correct, and what's more, they've got some seriously elegant proofs.
And from what I recall, they do have some diagrams (e.g., their commutative algebra book).

Better books:Walter Rudin: Complex and Real Analysis (splendid book for students, great explanation of measure theory),functional analysis from same author is good too.Kato: Perturbation theory (very good book but quite hard, wait till graduate).Brezis: functional analysis (french): a little abstract but short. Some early chapters are a little boring.Yosida: functional analysis (haven't had time to finish it, so my opinion on it is not set).Hormander: Series on linear operators are a reference. Very long.Ad

yea, but how many of those authors published books in the 1930's? unless Walter Rudin started publishing his writings in his early-teens, and Hormander before he turned 10, i don't think you can make a comparison between their works and Bourbaki's.

I think perhaps you weren't born to be a mathematician. I seriously wish that some of my profs would have STFU'd about the applications and focused on the proofs and proof techniques. If you are studying biology, do you really want half your class hours devoted to what plants look nice in a garden? (Maybe you do, but if so, study gardening instead.) If you are studying software architecture, should the textbook assume that what you really care about is, I don't know, writing keyboard scanners? (Again, you might, but then why not buy a different book?) And do you want your general psych class turned into a course on methods of military indoctrination? It doesn't matter the field, I think we'd benefit from a lot less focus on applications and a lot more on mastery of content. Mathematics most of all, because the cultural content of mathematics is the collection of tools for thinking about pure problems, abstracted from any problem domain. Indeed, the best advances in mathematics come, it seems to me, from abstracting internal mathematical tools away from their original mathematical focus, and thereby making them available to the whole subject, and not just one small field.

As to issues with how theorems are referred to, I think this brings us to the root of the Bourbaki phenomenon. The cult of personality is not so productive in a field whose content is supposedly objective, and naming results after people is a barrier to objectivity in understanding and a barrier to communication. My girlfriend is Chinese. Do you suppose she knows, or cares, about Green's theorem or Taylor series, under those names? But five seconds with a pencil and paper and we are in sync.

Mathematics is not automotive mechanics and it is not pop music! And—I don't mean to be rude here, in making a cultural observation—that was a particularly hard lesson for French academia in particular—though for France we needed to write "it is not the civil service and it is not religion."

The idea that career choices are predetermined at birth is a popular romantic view (cf. the human literary corpus of epics and fairy tales about Chosen Ones), but there is essentially no hard evidence for its validity, and I think it devalues the richness and variability of life experiences. Also, I don't think we should exclude people from mathematics just because they don't like the sort of dry abstraction you find in Bourbaki texts. There are plenty of reasonably successful mathematicians who are more comfortable learning things when they have an example or application in mind. For example, Timothy Gowers [wikipedia.org] wrote two [wordpress.com] posts [wordpress.com] on his blog, suggesting that exposition is improved by starting with examples to motivate an idea.

It doesn't matter the field, I think we'd benefit from a lot less focus on applications and a lot more on mastery of content. Mathematics most of all, because the cultural content of mathematics is the collection of tools for thinking about pure problems, abstracted from any problem domain.

From a practical standpoint, I don't think we should try to change teaching methodology too much at a time, because there are almost always weaknesses in revolutionary plans that don't show up at the thought experiment stage. More abstractly, I think people tend to learn content better when it is motivated with a useful context. Exactly where this balance should be struck is still a contentious issue (see math wars [wikipedia.org]), but I don't think Bourbaki is the answer. Even among pure academics, we value theoretical work by some notion of applicability. We say that etale cohomology is a good theory, not because it lets you think abstractly about pure stuff (although it does), but because you can use it to prove hard quantitative statements like the Weil conjectures, the Adams conjecture, and many theorems in representation theory.

I don't, in fact, disagree with you very intensely, and yes, perhaps I spoke too strongly. I do think that there is a real current trend away from teaching subjects according to their own self-determined core values and towards what politicians and industrialists would like, and I truly believe this to be short-sighted, damaging, and not particularly justified by education theory, historical experience, or much else. That's not quite the same as me wanting to launch into untested didactic waters without a l

The principal drawback of the Bourbaki school is not that its output is "boring", but that it presents the final results of a theorem in a pristine format while removing all references to the messy drudgework and dead ends it took to get there. As such, it's pleasingly elegant for practicing mathematicians, but terrible for students. Students need to see examples of math as a process, not just as a finished product.

Indeed. As far as that goes, I'm having a lot of fun with the book "Numbers and Proofs". It mostly focuses on number theory, but is primarily about the process of proving, and proof techniques. It's not a heavy duty serious textbook for university level work, but it's a nice book pre-university (which is where I am).

From what I can tell, Metascore is an attempt by mathematicians to take over the government. In fact, every government.http://www.metascore.org/ [metascore.org]Difference is they do not seem to be very secretive.

I don't want to get off on a tangent, but the whole group seem kind of irrational.
How does someone become a member of this finite group? Do they have to stand in the middle while the rest of them form a perfect square around them? I wonder if they have to hide their identity?

Just a quick comment from the left field (mama never taught me to be discrete): maybe they could stand in a ring instead (of a square) during their analysis of these complex issues, during their attempt to see what can be derived from the facts. That would seem more natural and rational; at least it's the norm.

The result was austere books with almost no examples, guide for intuition or pictures. Philip Davis of Brown University described them in an article in SIAM News as "mathematics with all its juices extracted; bare bones, skeletonic, anorexic stuff; Twiggy dressed in the tunic of Euclid." Michael Atiyah of the University of Edinburgh says: "They're not designed to be read. They're designed to set out a the is for how mathematics ought to be done."

Any they thought other books were dry? I guess books like this may have some use for hard core math types, but they sound like horrible books for almost anyone else. Examples, pictures, and the likes are very important for learning. Designing books not to be read seems like silly exercise.

Bourbaki was writing for a graduate student or professional level mathematician, not a public audience. The writing is dry by today's standards - indeed, many texts today use more prose and include diagrams. However, Bourbaki was very good at getting the mathematics itself clearly defined.

However, Bourbaki was very good at getting the mathematics itself clearly defined.

It's just a pity they were never able to clearly get it across.

Bourbaki, and the Bourbaki style, makes great reference material. But that's all it makes. There is more to mathematics, and pictures and example are part of that "more". A big part. Bourbaki did not just forget these topics. They actively excluded them. Jean Dieudonne [wikipedia.org] stood up in the middle of a conference and shouted "Down with Euclid! Death to Triangles!". It wa

Many people blame Bourbaki for the horrendous "new math" which infected mathematics teaching in the 1960's. And there is some validity to that accusation. A scathing indictment of Bourbaki was given by the renowned mathematician V.I. Arnold, author of famous books on classical mechanics and differential equations. Arnold tears apart the dry, lifeless and phony "rigor" and "purity" of Bourbaki and others who divorce mathematics from reality, which he describes as "sectarianism and isolationism which destroy

You might not be French. I am. And pure mathematicians here don't not care about applications. They actively deny their work has any purpose. For philosophical reasons, they see any application of mathematics as dirty. "Pure" mathematicians" and "applied" mathematicians actually don't talk to each other here. This is even more surprising once you learn that what is considered "applied mathematics" here is just considered "pure mathematics" elsewhere.

Aargh! From these mathematicians grew the "New Math" of the early 1960's.

During the 1950's, high school math was mainly geometry, algebra, trig, and calculus.

Then came the New Math. Imported from France, it emphasized set theory, number bases, and abstract number theory. Students learned cardinality, commutative laws, associative laws, and "pure" math, with less applied math and problem-solving.

Many educators (and even more parents) saw the New Math as being too abstract for daily use and undercutting concrete skills such as computation. Physicists, especially objected, when college freshmen could calculate in multiple number bases, but couldn't solve algebraic equations.

Mathematician/singer Tom Lehrer wrote the song,"New Math", with the line, "It's so very simple that only a child can do it!"

One book, Why Johnny Can't Add - the Failure of the New Math, pointed out that in the mid 1970's teachers applauded the death of the New Math. By the late 70's, algebra was back in style, and even trigonometry was being taught. So ended the French invasion of high school math classes.

The latest, of course, is the new-new-math, also called rain-forest math. Don't get me started...

"I survived four years of New Math - it was so easy that you couldn't do anything wrong. Straight A's in math..... Millions of adults are now math-illerates because of these oh-so-pure mathematicians."

I agree but I think the whole problem is that we don' start with a geometric interpretation of math, the mayan's used shapes for numerals, and arabic symbols hide mathematical truths that are expressed better in images, visual geometric shapes.

I had new math and as a result my equation solving ability was utterly horrible and honestly still is... but I did find that new math laid the groundwork for databases down the road... cardinality and set theory are sort of the gist of databases... and, well, commutative and associativity are useful for understanding operators in programming languages.

Did they actually push "New Math." As you said it was imported. They were writing to college audiences. This stinks of jumping on the band wagon. Someone heard how great this was from publisher's salesmen and demanded kids learn it.

I don't know at the books (listed at the parent's link) but the journal 'Seminaire N. Bourbaki' is at NUMDAM [numdam.org]. I'm not sure about the formal relationship but much of the journal is in a fairly similar style to the books. There's lots of other good French mathematics at the same site.

They can make RTFA about mathematicians to sound like this is an article about some couturier. Well.. It's all geometry after all!.. AND applied to nude models;) so this "secret society" should be into something, eh

Yeah.. This is nice anecdote playing on The Matrix (Hugo Weaving playing agent Smith), Lord of the Rings (Hugo Weaving playing Elrond, father of Arwen played by Liv Tyler) and who Liv Tyler's father is:)

1. Bourbaki was not a "secret society", although it didn't publish a membership list because membership wasn't particularly official. Even some non-French mathematicians participated occasionally.2. They did not publish "text" books, they published carefully written reference books.3. The reputation was they they met once a year or so in a nice French resort with a reputation for good wine, to enjoy themselves and argue about the best wording for proofs in the next volumes.4. While in principle, you might r

Oh, Anonymous Coward, how I love your posts. The complete disregard to logic, spelling and grammar appearing in every topic has become my warm security blanket. You are the nougat center of my candybar and I will henceforth use random caps keys while starting all sentences with a lowercase letter - or better an ampersand - as a show of the esteem I hold you in.

& sINCE yOU post this in every topic perhaps you might want to edit the oRIGINAL.